This is not really an e-book, but comes quite close. The full text is available on its author's web page along with a list of errata and even the MathSciNet review. One might expect that this could initially make it less likely for some to purchase the hard copy, especially graduate students living on a tight budget. However, if the author's intention is to be read, then this is a very good tactic. Those interested in Coxeter groups will have the opportunity to take a serious look into the book and, if my hunch is correct, will be quite excited by what they see. They may even get so thrilled that they will go ahead and buy the beautiful hardcover. I think more mathematician-authors should consider making their manuscripts (at least partially) available online; I salute Michael Davis for doing it (and his publisher for going along with it).

Now to the book itself.

Coxeter groups are ubiquitous in modern mathematics. There are therefore several books on the subject in various flavors catering to readers with differing priorities. A representation theorist's choice will most likely be Humphreys' Reflection Groups and Coxeter Groups, while a combinatorialist will probably go for Björner and Brenti's Combinatorics of Coxeter Groups. Michael Davis has written the one appropriate for geometric group theorists (finally!). Though this is undeniably a research monograph that focuses mainly on the geometric group theory point of view, I can easily see the first five chapters of it being a useful introduction for anyone interested in Coxeter groups.

The first chapter is an introduction to the whole text, and therefore can be skimmed through at a first reading, especially by readers who are looking to gain an insight into the main focus of the text but are not yet ready to delve into the details. Chapter 2 is a sufficiently comprehensive exposition of the basic ideas of geometric group theory that will be necessary in the rest of the book. The next two chapters provide the foundational definitions, examples and (combinatorial) constructions necessary for any study of Coxeter groups, and Chapter 5 introduces the basic geometric realization of a given Coxeter system. Thus the reader who comes to Chapter 6 has already been exposed to both the combinatorial and geometric approaches.

This can be a good stopping point for those who were merely looking for an introduction to Coxeter groups. Reading only this far would still be a good use of one's time. However for those who want to learn more about geometric group theory and the role Coxeter groups play in it, leading up to and including most recent research in the area, the rest of the book will also prove to be a well-chosen read. The fact that at that point the text begins to read more like a research monograph will definitely be acceptable to such a reader. The author has personally contributed to several of the advanced ideas and techniques developed in these latter chapters, and he does a great job bringing all of the relevant strands together in this volume.

Readers who want to know more about the contents should definitely look into Ralf Gramlich's excellent MathSciNet review, which as you already read above is available on the author's web page. I will simply add a comment on the appendices: The book comes with ten appendices which make up almost one third of the book. These vary in difficulty but any reader will find at least one of them very useful. The topics covered in the appendices range from background material (e.g., cell complexes in Appendix A, regular polytopes in Appendix B, complexes of groups in Appendix E, homology and cohomology of groups in Appendix F) to material that extends what is already covered in the main body of the text (eg. the classification of spherical and Euclidean Coxeter groups in Appendix C, the geometric representation of a Coxeter group in Appendix D). Overall these appendices are an invaluable component of this book, and make it even more useful for students.

Gizem Karaali is assistant professor of Mathematics at Pomona College.

Chapter 8: THE ALGEBRAIC TOPOLOGY OF U AND OF ∑ 136
8.1 The Homology of U 137
8.2 Acyclicity Conditions 140
8.3 Cohomology with Compact Supports 146
8.4 The Case Where X Is a General Space 150
8.5 Cohomology with Group Ring Coefficients 152
8.6 Background on the Ends of a Group 157
8.7 The Ends of W 159
8.8 Splittings of Coxeter Groups 160
8.9 Cohomology of Normalizers of Spherical Special Subgroups 163

Chapter 9: THE FUNDAMENTAL GROUP AND THE FUNDAMENTAL
GROUP AT INFINITY 166
9.1 The Fundamental Group of U 166
9.2 What Is ∑ Simply Connected at Infinity? 170

Chapter 11: THE REFLECTION GROUP TRICK 212
11.1 The First Version of the Trick 212
11.2 Examples of Fundamental Groups of Closed Aspherical
Manifolds 215
11.3 Nonsmoothable Aspherical Manifolds 216
11.4 The Borel Conjecture and the PDn-Group Conjecture 217
11.5 The Second Version of the Trick 220
11.6 The Bestvina-Brady Examples 222
11.7 The Equivariant Reflection Group Trick 225